## Representation theory – Connected components of real points regular semisimpler Place of the adjoint quotient of the real split-lie algebra are contractible

I am looking for a reference for the following question:

To let $$mathfrak {g}$$ be a split semisimple couch over algebra $$mathbb {R}$$ with corresponding split semisimple adjoint group $$G$$, Write $$B = mathfrak {g} : // , G$$ for the adjoint quotient a variety is isomorphic too $$mathbb {A} ^ { text {rank} (G)}$$ parameterization $$G$$invariant polynomials of $$mathfrak {g}$$, Write $$B ^ {rs} subset B$$ for the open subscheme of the regular semisimple elements the complement of the discriminant polynomial.
To let $$mathfrak {h} subset mathfrak {g}$$ be a Cartan subalgebra that does not necessarily have to be shared, and write $$mathfrak {h} ^ {rs} subset mathfrak {h}$$ for the regular semisimple locus of $$mathfrak {h}$$,
Since the morphism $$mathfrak {h} ^ {rs} rightarrow B ^ {rs}$$ is finally, the corresponding morphism $$pi: mathfrak {h} ^ {rs} ( mathbb {R}) rightarrow B ^ {rs} ( mathbb {R})$$ at real points (with the Euclidean topology) there will be a local homeomorphism with finite fibers. If $$U subset B ^ {rs} ( mathbb {R})$$ is then either a connected component $$pi ^ {- 1} (U)$$ is empty of $$pi ^ {- 1} (U) rightarrow U$$ is a cover room.

Question: is $$pi ^ {- 1} (U) rightarrow U$$ a trivial cover space?

That's true, though $$mathfrak {h}$$ is split because in this case the Weyl group acts transitive on the connected components of $$mathfrak {h} ^ {rs} ( mathbb {R})$$, Another way of formulating my question is whether it is true that the true points of the Weyl group of $$mathfrak {h}$$ acts transitive on the connected components of $$mathfrak {h} ^ {rs} ( mathbb {R})$$,
One solution would be to show that every connected component of $$B ^ {rs} ( mathbb {R})$$ is capable of contracting, but could not prove it. Every help is appreciated!

## Commutative algebra – dimension of a given finite generated quotient module over a local ring.

I have dealt with the following question from the theory of dimension in commutative algebra.

To let $$(A, m)$$ be a local ring and $$M$$ a finally generated $$A$$-Module.

given $$x_1, …, x_r in M ​​$$, Prove that $$dim ( frac {M} {(x_1, …, x_r) M}) geq dim (M) – r$$with equality, if and only if {$$x_1, …, x_r$$} is part of a parameter system for $$M$$,

Now I can show that, though $$A$$ is a $$mathbf {regular}$$ local ring so $$frac {A} {(x_1, …, x_r) A}$$ is a regular local ring with dimension $$dim (A) – r$$ then and only if {$$x_1, …, x_r$$} is part of a parameter system for $$A$$, But I do not know how to show that for the given case. I also can not show the inequality. I could not find any proof, so I would be grateful for any help!

## Abstract Algebra – Quotient of a free group on the compressible substrings of a string.

To let $$s$$ be a string over an alphabet $$Sigma$$, say $$s = abcdabcdab$$,

We write $$t leqslant s$$ mean $$t$$ is a substring of $$s$$ or is there $$u, v$$ Strings over $$Sigma$$ so that $$utv = s$$ where the multiplication is the concatenation of strings.

Define $$C (s) = {t leqslant s: tvt leqslant s,$$ for some $$v leqslant s$$ and $$| t | geq 2 }$$, These are all sorts of compressible symbols $$s$$,

Define $$I (s) = {t in C (s): C (t) = varnothing }$$, These are all possible terminal rules that can occur during a CFG generation $${s }$$, For our example $$s$$ given above $$I (s) = {ab, bc, cd, da, abc, bcd, cda, dab, abcd, bcda, cdab }$$,

Now $$i bullet i$$ to the $$i in I (s)$$ that represents at least two $$i$$can fit in $$s$$ by definition of $$C (s)$$,

To let $$n_s (t)$$ equal to the maximum number of $$t$$is depressible in $$s$$, You can calculate it by packing on the far left $$t$$ in $$s$$and it takes polynomial time. Calculation of $$I (s)$$, Polynomial in $$| s |$$,

Well, if $$i bullet cdots bullet i$$ ($$n_s (t) + 1$$ times) is set equal to $$1$$ a special identity symbol.

Now form the $$text {FreeGroup} (I (s)) =: F$$ and look at all the elements $$x = y ^ {- 1} zy$$ for some $$y in F$$ and $$z$$ that is impossible to have $$s$$: Then the amount of all these elements $$H$$ should form a normal subgroup.

$$z = i bullet i$$. $$i = from$$, Then, when $$s = abab$$. $$z bullet z = i bullet i bullet i bullet i$$ is not possible in $$s$$,

See what I'm trying to achieve? How can I make this construction more stringent?

## Group Theory – Canonical writing of a transitive action as a quotient of a simply transitive action

Consider a finite group $$G$$ Transitive effect on a finite amount $$Y$$, Is it possible to find a finite amount? $$P_Y$$ and a finite group $$has G$$ Impact on $$P_Y$$ so that the following is true?

• The action of $$has G$$ on $$P_Y$$ is regular, d. H. both free and transitive.
• There is a surjective map $$pi colon P_Y to Y$$ and a surjective group homomorphism $$phi colon has G to G$$ so that $$pi (gp) = phi (g) pi (p)$$ for each $$g in has$$ And everybody $$p in P_Y$$,
• The construction of $$P_Y$$. $$has G$$. $$pi$$ and $$phi$$ is & # 39; canonical / functorial & # 39; in the sense that, for example, it does not rely on the a priori choice of an otherwise indistinguishable element of $$G$$ Or from $$Y$$,

Note that without the last condition, the question would be trivial, as one could simply set any "origin". $$o in Y$$ and then take $$P_Y = has G = G$$. $$phi = id$$ and $$pi (h) = ho$$,

## Ag.algebraic geometry – Galois covering, which corresponds to the finite quotient of the étale basic group

To let $$X$$ be a connected scheme,$$pi_1 (X, bar {x})$$ his étale fundamental group for a geometric point $$bar {x}: Spec (K) rightarrow X$$
and $$E = pi_1 (X, bar {x}) / N$$ a finite quotient of $$pi_1 (X, bar {x})$$

I'm looking for a book or a work that describes the explicit structure of the Galois cover $$Y rightarrow X$$ corresponding to $$E$$ unlike the book by Grothendieck's SGA or Tamas Szamuely

## Agal Algebraic Geometry – Is it possible to embed a group schema into a locally constant one so that the quotient exists?

To let $$S$$ be a sufficiently good basic scheme (such as the finite type over an algebraic closed field) and $$G to S$$ be a flat group scheme. I would like to ask: Can we always find a closed embedding? $$G to H$$ into another flat group scheme $$H$$, so that $$H$$ is constant over S locally in the Zariski topology, and that $$H / G$$ exists as a schema?

## ag.algebraic geometry – What is the geometric quotient of the abelian triple number?

Consider a finite field $$mathbb {F} _p$$ so that $$p equiv 1 ( mathrm {mod} 3)$$ and his element $$zeta neq 1$$. $$zeta ^ 3 = 1$$,

Also let $$E !: Y ^ 2 = x ^ 3 + b$$ be an elliptic curve of $$j$$-invariant $$0$$, from where $$b in mathbb {F} _p ^ * setminus ( mathbb {F} _p ^ *) ^ 3.$$ This curve has the order $$3$$ automorphism $$( zeta) !: (x, y) mapsto ( zeta x, y).$$

Consider the diagonal matrices
$$A: = mathrm {diag} (1, zeta, zeta ^ 2), qquad B: = mathrm {diag} ( zeta ^ 2, zeta, 1).$$
They form the subgroup $$G: = langle A, B rangle subset mathrm {SL} (3, mathbb {Z} ( zeta)),$$ That's isomorphic too $$( mathbb {Z} / 3) ^ 2$$, Any element $$mathrm {diag} ( alpha, beta, gamma) in G$$ of course works on the triple abelsche $$E ^ 3$$:
$$(P, Q, R) mapsto ((α) P, (β) Q, (γ) R).$$

What is the geometric quotient? $$E ^ 3 ! / G$$? Could you explicitly write an (affine) model for this variety?

## List Manipulation – Grassman and Clifford algebras as quotient of tensor algebra

I worked with tensors in Mathematica. It's great to have TensorProduct with linearity and everything. Perhaps you can use these structures to work with Grassman algebras or Clifford algebras, both of which are quotient of tensor algebra: T / v ^ 2 is the Grassman algebra, T / (v ^ 2 + norm (v)) the Clifford algebra algebra.

After thinking about it for three days, I still do not know exactly how to switch from tensors to Grassman / Clifford.

## real analysis – via quotient space and linear map

I'm reading a book called Functional Analysis written by Peter D Lax

Definition M is a linear map of$$X rightarrow U$$.$$N_M$$ is the set of points that are mapped to zero.
$$R_M$$ is the range of M, is the image of X under M in U

He wants me to prove that M maps the quotient space $$X / N_M$$ one on one $$R_M$$

Here is my confusion first, I think the element of the quotient space is like $$y_1, y_2$$ ,so that $$y_1-y_2 in N_M$$So M (y_1-y_2) = 0, then M (y_1) = M (y_n). But that contradicts one on one. Where is my mistake? I do not know how to make mistakes

## Abstract Algebra – Herbrand Quotient

I'm trying to solve exercises from Lang's algebra and stick to a problem with Herbert quotients.

To let $$G$$ Let be a finite cyclic group of orders $$n$$ generated by an element $$sigma$$, Accept that $$G$$ works in an abelian group $$A$$, and let $$f, g: A rightarrow A$$ are the endomorphisms of $$A$$ given by $$f (x) = sigma x – x$$ and $$g (x) = x + sigma x + ldots + sigma ^ {n -1} x$$,

Define the Herbrand quotient by the expression $$q (A) = (A_f: A ^ g) / (A_g: A ^ f)$$assuming both indexes are finite. Now let us assume that $$B$$ is a subgroup of $$A$$ so that $$G B subseteq B$$,

(a) Naturally define an operation of $$G$$ on $$A / B$$,

(b) Prove that $$q (A) = q (B) q (A / B)$$
in the sense that if two of these quotients are finite, then the third and the specified equality holds.

(c) If $$A$$ is finally, show that $$q (A) = 1$$,

The $$A_f, A ^ f$$ in question is the image and the core of $$f$$ as a map of $$A$$ to $$A$$, The same applies to $$B_f$$ and $$B g$$,

I'm stuck right now $$(b)$$ and I do not understand why $$f, g$$ are endomorphisms of groups at the beginning. If $$G$$ affects $$A$$It is not necessary for everyone $$g in G$$ induces an automorphism $$A$$ Law? Thank you in advance!